Vector-valued Reproducing Kernel Banach Spaces with Group Lasso Norms

Abstract

Focusing on establishing a mathematical basis for kernel methods in sparse multi-task learning, we explore the theory of vector-valued reproducing kernel Banach spaces (RKBSs) endowed with $\ell_{p,1}$-norms ($1\le p\le +\infty$), encompassing both the sparse learning case when $p=1$ and the group lasso when $p=2$. We develop RKBSs equipped with these group lasso norms that support the linear representer theorem for regularized learning frameworks. Additionally, we introduce reproducing kernels admissible for this construction. Such reproducing kernels are applicable to sparse multi-task learning with group lasso norms.